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```1914]    SOME  CALCULATIONS IN ILLUSTRATION  OF  FOURIER'S THEOREM     231
An inspection of the curves plotted from the above tables shows the approximation towards discontinuity as ^ increases.
That the curve remains undulatory is a consequence of the sudden stoppage of the integration at & = &a. If we are content with a partial suppression only of the shorter wave-lengths, a much simpler solution is open to us. We have only to introduce into (1) the factor e~ak., where a is positive, and to continue the integration up to as = oo . In place of (2), we have
tan-1
<f> (a,) = rdl™~~ {Sin % 0 + 1) _ sin k(x- 1)1 = tan"1 (-—} -Jo      «                                                           \   a   J
...... (9)
The discontinuous expression corresponds, of course, to a = 0. If a is merely small, the discontinuity is eased off. The following are values of \$(#), calculated from (9) for a = 1, 0'5, 005 :
X	\$(x)	X	*(*)	X	</>(")
o-o	1 -571	2-0	0-464	4-0	0-124
0-5	1-446	2-5	0-309	o-o	O'OSO
1-0	1-107	3-0	0-219	c-o	0-OB5
1-5	0-727
a = 0-5.
a = 0-05.
X	<!> («)	X	•HT)	X	</> (»)
o-oo	2-214	1-00	1-326	2-00	0-298
0-25	2-J73	1-25	0-888	2-50	0-180
0-50	2-111	1-50	0-588	3-00	0-120
0-75	1-756	1-75	0-408	3-50	0-087
X	*(*)	i X	*(*)	x	0(*)
o-oo	3-041	0-90	2-652	1-20	0-222
0-20	3-037	0-95	2-331	1-40	0-103
0-40	3-023	1-00	1 -546	1-60	0-064
0-60	2-986	1-05	0-761	1-80	0-045
0-80	2-869	1-10	0-440	2-00	0-033
As is evident from the form of (9), <£ (as) falls continuously as as increases whatever may be the value of a.                       .     ' .59525		45	1-55871	56	1-55574
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